Take a LKT \(\mu \)  of \(\pi \) . Then its infinitesimal character has the form (REF ). By the Dirac inequality (REF ), one has that
\(\Vert \Lambda \Vert ^2=\Vert \lambda _a(\mu )\Vert ^2 +\Vert \nu \Vert ^2\le \Vert \mu \Vert ^2_{\rm spin}.\) 
Therefore,
\(\Vert \nu \Vert ^2\le \Vert \mu \Vert ^2_{\rm spin}- \Vert \mu \Vert ^2_{\rm lambda}.\) 
As computed in Section 5.8 of [1]}, we have that \(\max \lbrace A_j\mid 0\le j\le 62\rbrace =82\) . We refer the reader to Section 3 of [1]} for the precise meaning of \(A_j\) . It follows that for any u-large \(K\) -type \(\mu \) , one has that
\(\Vert \mu \Vert _{\rm spin}^2 - \Vert \mu \Vert _{\rm lambda}^2\le 82.\) 
Since \(\Vert \nu \Vert \ge \sqrt{82.5}\) , we conclude from (REF ) that \(\mu \)  can not be u-large.
There are 53709 u-small \(K\) -types in total. Among them, only 23 have the property that
\(82.5\le \Vert \mu \Vert _{\rm spin}^2 - \Vert \mu \Vert _{\rm lambda}^2.\) 
Let us collect these 23 u-small \(K\) -types as Certs. Its elements are listed as follows:
\(&[0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 6], [0, 0, 0, 0, 0, 1, 3],\\&[0, 0, 0, 0, 0, 1, 5], [0, 0, 0, 0, 0, 1, 7], [0, 0, 0, 0, 0, 2, 4], [0, 0, 0, 0, 0, 2, 6], \\&[0, 0, 0, 0, 0, 3, 7], [0, 0, 0, 0, 1, 0, 2], [0, 0, 0, 0, 1, 0, 4], [0, 0, 0, 0, 1, 0, 6], \\&[0, 0, 0, 0, 1, 1, 3], [0, 0, 0, 0, 1, 1, 5], [0, 0, 0, 1, 0, 0, 6], [0, 0, 1, 0, 0, 0, 5], \\&[0, 1, 0, 0, 0, 0, 4], [0, 1, 0, 0, 0, 0, 6], [1, 0, 0, 0, 0, 0, 3], [1, 0, 0, 0, 0, 0, 5], \\&[1, 0, 0, 0, 0, 0, 7], [1, 0, 0, 0, 0, 1, 4], [1, 0, 0, 0, 0, 1, 6].\) 

Moreover, we compute that
\(1.5\le \Vert \lambda _a(\mu )\Vert ^2\le 26.5\)  for any \(\mu \in \texttt {Certs}\) . Therefore,
\(1.5+ 82.5 \le \Vert \Lambda \Vert ^2=\Vert \lambda _a(\mu )\Vert ^2+\Vert \nu \Vert ^2\le 26.5 +113.5.\) 
The right hand side above uses the proven bound (REF ). There are 1113 integral \(\Lambda \) s meeting the requirement (REF ). We collect them as \(\Omega \) .
Now a direct search using atlas says that there are 1754 irreducible representations \(\pi \)  such that \(\Lambda \in \Omega \)  and that \(\pi \)  has a LKT which is a member of Certs. Furthermore, only one of them turns out to be unitary: the minimal representation. This finishes the proof.
